1. Further Investigations on Averages

2. Let’s consider the data set 1, 2, 4, 6, 6:
Mean
Median
Mode
Original data set
1, 2, 4, 6, 6
3.8 4 6
New data set (add 5 to each datum in the original set)
8.8 9 11
6, 7, 9, 11, 11
( )
(4 + 5)
(6 + 5)
New data set (multiply each datum in the original set by 2)
7.6 8 12
2, 4, 8, 12, 12
(3.8  2)
(4  2)
(6  2)

3. From the previous page, we can summarize the properties of averages as follows:
Adding a constant C to each datum:
New arithmetic mean = original arithmetic mean + C
New median = original median + C
New mode = original mode + C
Multiplying a constant k to each datum:
New arithmetic mean = original arithmetic mean  k
New median = original median  k
New mode = original mode  k

4. Consider another data set 1, 2, 4, 5, 6, 6:
Mean
Median
Mode
Original data set
1, 2, 4, 5, 6, 6 4
4.5 6
New data set (insert a datum ‘0’ in the original set)
3.167 3 6
0, 1, 2, 4, 5, 6, 6
(< 4)
(< 4.5)
(= 6)
New data set (delete a datum ‘2’ in the original set)
4.4 5 6
1, 4, 5, 6, 6
(> 4)
(> 4.5)
(= 6)
New data set (delete a datum ‘5’ in the original set)
3.8 4 6
1, 2, 4, 6, 6
(< 4)
(< 4.5)
(= 6)

5. From the previous page, we can summarize the properties of averages as follows:
Inserting the datum ‘0’:
For a set of data with positive values only, if the datum ‘0’ is inserted:
The arithmetic mean will decrease.
The median will decrease or remain unchanged.
The mode will not be changed.

6. Deleting a datum:
If a datum is deleted from a set of data:
The arithmetic mean will increase (decrease) if the datum deleted is smaller than (greater than) the arithmetic mean.
In general, the median will increase (decrease) or remain unchanged if the datum deleted is smaller than (greater than) the median.
In general, the mode will not be changed if the datum deleted is not the mode.

7. Follow-up question
Consider a data set: 4, 8, 12, 28, 28, 34. (a) Find the arithmetic mean, the median and the mode of above data. (b) After 4 is added to each of the data, find the new arithmetic mean, median and mode. (c) If the datum ‘4’ is deleted from the data set, find the new arithmetic mean, median and mode.

8. Follow-up question (cont’d)
Consider a data set: 4, 8, 12, 28, 28, 34. (a) Find the arithmetic mean, the median and the mode of above data.
Solution 6 34 28 12 8 4 + =
(a) Arithmetic mean 19 = 2 28 12 + =
Median 20 =
Mode 28 =

9. Follow-up question (cont’d)
Consider a data set: 4, 8, 12, 28, 28, 34. (b) After 4 is added to each of the data, find the new arithmetic mean, median and mode.
Solution
(b) Arithmetic mean 4 19 + = 23 =
Median 4 20 + = 24 =
Mode 4 28 + = 32 =

10. Follow-up question (cont’d)
Consider a data set: 4, 8, 12, 28, 28, 34. (c) If the datum ‘4’ is deleted from the data set, find the new arithmetic mean, median and mode.
Solution 5 34 28 12 8 + =
(c) Arithmetic mean 22 =
Median 28 =
Mode 28 =